Neil Tennant wrote on Mon, 12 Oct 1998 11:44:37
> ...
>> I asked Sazonov to undertake a thought experiment: "ask yourself
> what theory of arithmetic would be excogitable by a disembodied
> Cartesian soul in a universe with no physical objects in it at all."
> To this he replied "Unfortunately, I am not inclined to make such kind
> of experiments. (Probably this is the main point where we disagree.)"
>> In other words, he is saying "I dare not go there; I fear I shall be
> led by the nose to the horrible realization that my theory is really
> rather fatuous after all."
>> Compare these analogous cases, where T is trying to get a stubborn S
> to reach a conclusion by making a vital thought experiment:
>> T: Ask yourself whether you would like to be treated the way Jews were
> treated in Nazi Germany.
> S: Unfortunately, I am not inclined to make such thought experiments.
> [Conclusion: S must be either evil or morally blind.]
>> T: Ask yourself whether there might not be a world in which some
> people behaved exactly as they do in this world, but in which they
> have no inner experiences---no sensations, no emotions etc.
> S: Unfortunately, I am not inclined to make such thought experiments.
> [Conclusion: S is blind to the problem of consciousness and other
> minds.]
>> T: Ask yourself whether there might not be a world in which all
> emeralds are green if examined before the year 2000 and blue otherwise.
> S: Unfortunately, I am not inclined to make such thought experiments.
> [Conclusion: S is blind to Goodman's new riddle of induction.]
>> I shall refrain from developing such a list ad nauseam.
>> Sazonov asks 'Also, what does it mean the "disembodied Cartesian soul"
> and a "universe with no physical objects"? This is again something
> mysterious. If there are no objects then there is nothing to count.'
>> Wrong!! Just because there are no *physical* objects does not entail
> that "there is nothing to count"! There are *the numbers themselves*
> (abstract objects) to count! That's what was happening when each
> natural number n was revealed as the number of preceding natural
> numbers. That, one wants to stress, was the whole point of the thought
> experiment. Little wonder that Sazonov is "not inclined" to engage in
> such a thought experiment.
And that is all? It is *this* your "vital" argument (with finite
ordinals in a set-theoretic universe without urelements =?
physical objects)? Then I do not know what is a scientific
argument at all. I have no words! When you asked me to proceed
(by my free will, isn't so?) in the direction you showed me
("disembodied Cartesian soul" and a "universe with no physical
objects"), I anticipated something of such a kind. I am really
not inclined to follow this way independently on your allusions
concerning my scientific *honesty* etc. The latter may be also
considered as you are actually trying to FORCE me to go this way
which, I am therefore FORCED to say not very politely, seems to
me having a doubtful quality.
You (as well as me and all of us), being a free person, may
entice everybody to go your way. But everybody has a right not
follow that way and manner of thinking.
I think it would be *honest* for you to agree that a concept
(like absolute mathematical truth or the unique mathematical
world, or the unique standard model of PA) does not have a
coherent meaning instead of making a violence on your own and
others mind to assert contrary.
It is well known to me that axioms of Peano arithmetic arise in
a very natural way. The main point is Induction Schema. It can
be understood, say, via its equivalent form as the Minimum
Principle: every definable nonempty *finite* set of numbers has
a minimal element. This immediately reflects our everyday
experience with considering finite segments of natural numbers
(or finite sequences of pebbles or the like).
Also note, that there is seemingly *nothing* in the Minimum
Principle which would say *a priori* that we are inevitably
formalizing also non-feasible numbers, i.e. numbers outside our
practical calculations. Thus, we could conclude in principle
that Peano Arithmetic is an ideal formalization of feasible
numbers. It is interesting to understand *how* is it possible to
get such a wrong conclusion by using seemingly evident and
inoffensive steps? What really happened and how, nevertheless,
to formalize feasible numbers more properly? At least for
"thought" experiment. Say, to find a serious philosophical
reason why it is impossible.
But if it is possible, then we would get as a bonus a new,
somewhat more realistic arithmetic with some
complexity-theoretic flavour which probably will have better
relations with practice and, IF we will be happy, will shed a
new light onto other branches of mathematics involving numbers
[say, what will happen with Analysis?] and on the problems of
complexity theory like P=?NP.
I think, there is no (and we actually need not it at all!)
mysticism in the above description of the situation with
formalizing (the ordinary or feasible) arithmetic. The only
not very realistic thing is some (not very clear in the
infinity) model of PA arising in our imagination (as it is
essentially in the case of non-Euclidean geometries from
the point of view of our everyday experience) essentially
due to first-order logic on which PA is based. But it is
nothing bad, even very helpful and actually inevitable to
use our imagination as much as possible if it is regulated
(and enormously strengthened!) by a formal system.
In which place of this description we need the mysterious
notion of "standard" or "unique" model of natural numbers
at all? I honestly say that the model in *my* imagination
is something *vague* thing, however the formalism PA which
it describes is sufficiently clear to use it for concrete
proofs of concrete theorems of PA.
[I will not present now any concrete arguments against absolute
believing in induction axiom. This have been done in some my
older postings. Cf. for ex. that from Fri, 09 Jan 1998 00:10.]
Taking axioms, each one separately plausible, does not mean that
all their consequences will be plausible in all respects as
well. The well-known example is Choice Axiom. It is quite
normal situation that after formalization of some idea we can
get something unexpected and unforeseen for which it would be
problematic to give intuitive explanation of the same kind as we
started. We should not overestimate the role of formalization
and should not consider any formal theory as something given to
us by the God. They are just specially constructed by peoples
*instruments* or *devices* of a specific kind which make our
thinking incomparably stronger than when thinking purely
informally (if this "purely" is possible at all!). Therefore
any such instrument may be changed or corrected by us, if
necessary.
Any instrument cannot be true or false (especially in any
absolute sense). It may be only more or less useful, convenient,
applicable and coherent with respect to other instruments. And
the notions of truth or falsity should be used rather carefully
to distinguish their purely *technical* meaning which is
implicit in formal rules of classical logic from their meaning
related with application or interpretation of a given formalism
in the real world.
> ...
Vladimir Sazonov